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Answer :
To solve this problem, we need to determine how long it takes for 300 grams of Element X to decay to 80 grams, given that its half-life is 11 minutes. We can use the formula for exponential decay, which is:
[tex]\[ y = a \times (0.5)^{\frac{t}{h}} \][/tex]
Where:
- [tex]\( y \)[/tex] is the remaining amount of the substance.
- [tex]\( a \)[/tex] is the initial amount of the substance.
- [tex]\( t \)[/tex] is the time that has passed.
- [tex]\( h \)[/tex] is the half-life of the substance.
In this case:
- The initial amount [tex]\( a = 300 \)[/tex] grams
- The remaining amount [tex]\( y = 80 \)[/tex] grams
- The half-life [tex]\( h = 11 \)[/tex] minutes
We need to find [tex]\( t \)[/tex], which is the time it takes for the substance to decay from 300 grams to 80 grams.
Here are the steps to find [tex]\( t \)[/tex]:
1. Set up the equation using the decay formula:
[tex]\[ 80 = 300 \times (0.5)^{\frac{t}{11}} \][/tex]
2. Divide both sides by 300 to isolate the decay term:
[tex]\[ \frac{80}{300} = (0.5)^{\frac{t}{11}} \][/tex]
3. Simplify [tex]\(\frac{80}{300}\)[/tex]:
[tex]\[ \frac{4}{15} \approx 0.2667 \][/tex]
4. To solve for [tex]\( t \)[/tex], take the logarithm of both sides. We can use the base 10 logarithm (common log) or natural logarithm (ln), but we'll use the natural logarithm in this case:
[tex]\[ \ln(0.2667) = \ln((0.5)^{\frac{t}{11}}) \][/tex]
5. Using the logarithmic identity [tex]\(\ln(a^b) = b \times \ln(a)\)[/tex], rewrite the equation:
[tex]\[ \ln(0.2667) = \frac{t}{11} \times \ln(0.5) \][/tex]
6. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = 11 \times \frac{\ln(0.2667)}{\ln(0.5)} \][/tex]
7. After calculating the above expression, you'll find that [tex]\( t \approx 21.0 \)[/tex].
Therefore, it will take approximately 21.0 minutes for Element X to decay from 300 grams to 80 grams.
[tex]\[ y = a \times (0.5)^{\frac{t}{h}} \][/tex]
Where:
- [tex]\( y \)[/tex] is the remaining amount of the substance.
- [tex]\( a \)[/tex] is the initial amount of the substance.
- [tex]\( t \)[/tex] is the time that has passed.
- [tex]\( h \)[/tex] is the half-life of the substance.
In this case:
- The initial amount [tex]\( a = 300 \)[/tex] grams
- The remaining amount [tex]\( y = 80 \)[/tex] grams
- The half-life [tex]\( h = 11 \)[/tex] minutes
We need to find [tex]\( t \)[/tex], which is the time it takes for the substance to decay from 300 grams to 80 grams.
Here are the steps to find [tex]\( t \)[/tex]:
1. Set up the equation using the decay formula:
[tex]\[ 80 = 300 \times (0.5)^{\frac{t}{11}} \][/tex]
2. Divide both sides by 300 to isolate the decay term:
[tex]\[ \frac{80}{300} = (0.5)^{\frac{t}{11}} \][/tex]
3. Simplify [tex]\(\frac{80}{300}\)[/tex]:
[tex]\[ \frac{4}{15} \approx 0.2667 \][/tex]
4. To solve for [tex]\( t \)[/tex], take the logarithm of both sides. We can use the base 10 logarithm (common log) or natural logarithm (ln), but we'll use the natural logarithm in this case:
[tex]\[ \ln(0.2667) = \ln((0.5)^{\frac{t}{11}}) \][/tex]
5. Using the logarithmic identity [tex]\(\ln(a^b) = b \times \ln(a)\)[/tex], rewrite the equation:
[tex]\[ \ln(0.2667) = \frac{t}{11} \times \ln(0.5) \][/tex]
6. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = 11 \times \frac{\ln(0.2667)}{\ln(0.5)} \][/tex]
7. After calculating the above expression, you'll find that [tex]\( t \approx 21.0 \)[/tex].
Therefore, it will take approximately 21.0 minutes for Element X to decay from 300 grams to 80 grams.
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